How to bevel a vertex corner that is between a circle and straight?
This pretty much exactly what the Spin tool is for .. 



In an orthogonal view of the circular arc you want to create, put the 3D cursor at the center of the new part-circle, select the vertex you want to extrude around it, and invoke the tool.

In your case, you might ShiftS snap the cursor to the two selected endpoints of the bevel, so it's exactly between them.

You can adjust the center during the operation by using the arrowheads, and the number of vertices generated in the sweep using the tool's Adjust Last Operation panel